Non-Local Thermo-Elastic Buckling Analysis of Multi-Layer Annular/Circular Nano-Plates Based on First and Third Order Shear Deformation Theories Using DQ Method
محورهای موضوعی : EngineeringSh Dastjerdi 1 , M Jabbarzadeh 2
1 - Department of Mechanical Engineering, Shahrood Branch, Islamic Azad University, Shahrood, Iran
2 - Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran
کلید واژه: Differential quadrature method (DQM), Multi-layer orthotropic annular/circular graphene sheets, Non-local first and third order shear deformation theories, Thermo-elastic buckling analysis,
چکیده مقاله :
In present study, thermo-elastic buckling analysis of multi-layer orthotropic annular/circular graphene sheets is investigated based on Eringen’s theory. The moderately thick and also thick nano-plates are considered. Using the non-local first and third order shear deformation theories, the governing equations are derived. The van der Waals interaction between the layers is simulated for multi-layer sheets. The stability governing equations are obtained according to the adjacent equilibrium estate method. The constitutive equations are solved by applying the differential quadrature method (DQM). Applying the differential quadrature method, the ordinary differential equations are transformed to algebraic equations. Then, the critical temperature is obtained. Since there is not any research in thermo-elastic buckling analysis of multi-layer graphene sheets, the results are validated with available single layer articles. The effects of non-local parameter, the values of van der Waals interaction between the layers, third to first order shear deformation theory analyses, non-local to local analyses, different values of Winkler and Pasternak elastic foundation and analysis of bi-layer and triple layer sheets are investigated. It is concluded that the critical temperature increases and tends to a constant value along the rise of van der Waals interaction between the layers.
[1] Boehm H.P., Setton R., Stumpp E., 1994, Nomenclature and terminology of graphite intercalation compounds, Pure and Applied Chemistry 66: 1893-1901.
[2] Novoselov K.S., Geim A.K., Morozov S.V., Jiang D., Zhang Y., Dubonos S.V., Grigorieva I.V., Firsov A.A., 2004, Electric field effect in atomically thin carbon films, Science 306: 666-669.
[3] Reddy J.N., 2011, Microstructure-dependent couple stress theories of functionally graded beams, Journal of the Mechanics and Physics of Solids 59(11): 2382-2399.
[4] Akgöz B., Civalek Ö., 2013, A size-dependent shear deformation beam model based on the strain gradient elasticity theory, International Journal of Engineering Science 70: 1-14.
[5] Akgöz B., Civalek Ö., 2013, Buckling analysis of functionally graded micro beams based on the strain gradient theory, Acta Mechanica 224: 2185-2201.
[6] Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P., 2003, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids 51(8): 1477-1508.
[7] Ke L.L., Yang J., Kitipornchai S., 2011, Free vibration of size dependent Mindlin micro plates based on the modified couple stress theory, Journal of Sound and Vibration 331: 94-106.
[8] Akgöz B., Civalek Ö., 2011, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro scaled beams, International Journal of Engineering Science 49: 1268-1280.
[9] Akgöz B., Civalek Ö., 2013, Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory, Composite Structures 98: 314-322.
[10] Yang F., Chong A.C.M., Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39: 2731-2743.
[11] Eringen A.C., Edelen D.G.B., 1972, On non-local elasticity, International Journal of Engineering Science 10: 233-248.
[12] Eringen A.C., 1983, On differential equations of non-local elasticity, solutions of screw dislocation, surface waves, Journal of Applied Physics 54: 4703-4710.
[13] Eringen A.C., 2002, Non-Local Continuum Field Theories, Springer-Verlag, New York.
[14] Eringen A.C., 2006, Non-local continuum mechanics based on distributions, International Journal of Engineering Science 44: 141-147.
[15] Wen C.C., Chang T.W., Kuo W.S., 2014, Experimental study on mechanism of buckling and Kink-band formation in graphene nanosheets, Applied Mechanics & Materials 710: 19-24.
[16] Ghorbanpour Arani A., Kolahchi R., Allahyari S.M.R., 2014, Non-local DQM for large amplitude vibration of annular boron nitride sheets on non-linear elastic medium, Journal of Solid Mechanics 6(4): 334-346.
[17] Mohammadi M., Goodarzi M., Ghayour M., AlivandS., 2012, Small scale effect on the vibration of orthotropic plates embedded in an elastic medium and under biaxial in-plane pre-load via non-local elasticity theory, Journal of Solid Mechanics 4(2): 128-143.
[18] Dastjerdi S., Jabbarzadeh M., Tahani M., Nonlinear bending analysis of sector graphene sheet embedded in elastic matrix based on nonlocal continuum mechanics, IJE Transaction B: Applications 28: 802-811.
[19] Anjomshoa A., Shahidi A.R., Shahidi S.H., Nahvi H., 2014, Frequency analysis of embedded orthotropic circular and elliptical micro/nano-plates using non-local variational principle, Journal of Solid Mechanics 7(1): 13-27.
[20] Pradhan S.C., 2009, Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory, Physics letters A 373: 4182-4188.
[21] He X.Q., Kitipornchai S., Liew K.M., 2005, Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction, Journal of the Mechanics and Physics of Solids 53: 303-326.
[22] Scarpa F., Adhikari S., Gil A.J., Remillat C., 2010, The bending of single layer graphene sheets: The lattice versus continuum approach, Nanotechnology 21(12): 125702.
[23] Samaei A.T., Abbasian S., Mirsayar M.M., 2011, Buckling analysis of a single layer graphene sheet embedded in an elastic medium based on nonlocal Mindlin plate theory, Mechanics Research Communications 38: 481-485.
[24] Zenkour A.M., Sobhy M., 2013, Nonlocal elasticity theory for thermal buckling of nanoplates lying on Winkler–Pasternak elastic substrate medium, Physica E 53: 251-259.
[25] Wang Y., Cui H.T., Li F.M., Kishimoto K., 2013, Thermal buckling of a nanoplate with small-scale effects, Acta Mechanica 224(6): 1299-1307.
[26] Pradhan S.C., Phadikar J.K., 2009, Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models, Physics Letters A 373: 1062-1069.
[27] Bellman R.E., Casti J., 1971, Differential quadrature and long-term integration, Journal of Mathematical Analysis & Applications 34: 235-238.
[28] Bellman R.E., Kashef B.G., Casti J., 1972, Differential quadrature: A technique for the rapid solution of nonlinear partial differential equation, Journal of Computational Physics 10: 40-52.
[29] Sepahi O., Forouzan M.R., Malekzadeh P., 2011, Thermal buckling and postbuckling analysis of functionally graded annular plates with temperature-dependent material properties, Materials and Design 32: 4030-4041.
[30] Wang C.M., Xiang Y., Kitipornchai S., Liew K.M., 1994, Buckling solutions for Mindlin plates of various shapes, Engineering Structures 16: 119-127.
